Lecture 2 Vector Algebra

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1、Outline•ScalarandVectorQuantitiesLecture2VectorAlgebra•VectorOperationsØVectorAdditionandSubtractionØProductsofVectorsØProductsofThreeVectors2.1ScalarandVectorThewidespreadacceptanceofvectorsin2.1.1ScalarelectromagneticfieldtheoryisdueinparttoAphysicalquantitythat

2、canbecompletelydescribedbyitsthefactthattheyprovidecompactmagnitudeiscalledascalar.(suchascharge,current,energy,mathematicalrepresentationsofcomplicatedmass,time,temperature,work)phenomenaandallowforeasyvisualization5tfx,yzt,,andmanipulation.4p[(x1)2(yz2)]2

3、2“简洁的语言是深奥理论的源泉”(唐诗、宋词)2.1.2VectorØAnyvectorcanbewrittenasthesumofitscomponentsintheAphysicalquantityhavingamagnitudeaswellasadirectionisthreeorthogonaldirections,asfollows:calledvector.(suchaselectricandmagneticintensities,force,velocity)rrAAiaiAjajAakkˆAvecto

4、rAcanbewrittenasvrrAAAaAˆAAaˆAArAAAxcosazAAzrWhereAisthemagnitude(andhastheunitanddimension)ofrAAycosbAAgAndaˆAisadimensionlessUnitVectorwithaunitymagnitudeAAzcosgabAyrAxOyhavingthedirectionofArrvvAAAA(aˆxcosaaayzcosbgcos)rvvvvxAaAˆaAaxcosaaayzcosbg

5、cosANote:aˆAisnotalwaysConstantVectoraAFigure1GraphicrepresentationofvectorA12.2VectorOperationsØPositionvector2.2.1VectorAdditionandSubtractionrrvM0vAB(AB)av(AB)avv()ABavvvvrRiiijjjkkkrxayazaxyzvrM0C=A+BAAØDistancevector（矢径）C=A+BABvvvBBRrrTwovector

6、sHead-to-tailruleParallelogramrulerrBAC=A-Brrr-BA-BABVectorsubtraction2)VectororCrossProduct2.2.2ProductsofVectorsApplication?1)ScalarorDotProductA´Ba@ABsinqnABA×B@ABcos(A,B)A×B=AiBi+AjBj+AkBkØTheproductisavectorperpendiculartotheplanecontainingAandB;itsØItisascal

7、ar，commutativeanddistributivemagnitudeisnumericallyequaltotheareaoftheparallelogram.A×B=B×AA×(B+C)=A×B+A×CA´BanØUnitvectorsrelationBaiaiajajakka1BBqABABsin(qAB)aaaaaa0anqABijjkkiBcosqABAAATheright-handruleØItisevidentthat•IllustratingthedotproductofAan

8、dB2A´B=-B´AnotcommutativeA×A=AØVectorproductobeysthedistributivelaw,ØTheexpressionofVectorproductintheorthogonalcurvilinearcoordinatesystem(u1,u2,u3):A´